There are two tests named after Wilcoxon.
The one you mention is the signed rank test, but there's also the rank sum test, which might well be suitable for your problem. It's sometimes called the Mann-Whitney test.
It can be used to test for a location shift, or a scale shift, or more general alternatives, including stochastic dominance. For more general differences still, you might consider a two-sample Kolmogorov-Smirnov test.
You might also consider a permutation/randomization test (particularly if you're interested in a difference in means).
(In R, both wilcoxon tests can be obtained via wilcox.test
, and the two Kolmogorov-Smirnov tests can be obtained via ks.test
. Randomization tests are also very easy to carry out.)